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From: Christopher Maloney <dude.domain.name.hidden>

Date: Fri, 16 Jul 1999 23:33:26 -0400

Higgo James wrote:

*>
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*> But even such an 'identical' human has different spatial co-ordinates and is
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*> therefore different, no?
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*>
*

Umm, no! "Space" is an emergent phenomenon as well. What could you

possibly mean that it has a different "spatial coordinate"?

*> > -----Original Message-----
*

*> > From: Christopher Maloney [SMTP:dude.domain.name.hidden]
*

*> > Sent: Wednesday, July 14, 1999 6:25 AM
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*> > To: everything-list
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*> > Subject: Re: Cardinality of the MW
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*> >
*

*> > Let me add my information to this confusing brew:
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*> >
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*> > hal.domain.name.hidden wrote:
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*> > >
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*> > > Russell Standish, <R.Standish.domain.name.hidden>, writes:
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*> > > > My memory is fading somewhat about transfinite cardinal
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*> > > > numbers. However, it seems to me that c \leq \aleph_1. \aleph_1 is the
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*> > > > cardinality of the set of all sets of cardinalilty \leq\aleph_0. Since
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*> > c
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*> > > > is the cardinality of the set of all subsets of N, which is a subset
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*> > > > of the set of all sets of cardinality \leq\aleph_0.
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*> >
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*> > This is wrong, from what I know. I agree with Hal below that aleph-1
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*> > is defined to be the "next" cardinal after aleph-0. That is, by
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*> > definition, there is no transfinite number with cardinality > aleph-0
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*> > and < aleph-1.
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*> >
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*> > > >
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*> > > > What has never been proven is that c=\aleph_1, although it is widely
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*> > > > suspected.
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*> >
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*> > In fact it has been shown that c == aleph-1 is not provable by the
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*> > axioms of Zermelo-Fraenkel set theory. This is known as the
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*> > continuum hypothesis (CH). CH is also not disprovable,
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*> > which means that it is independent of those axioms. Thus it is
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*> > possible to construct set theories which assume that ~CH, and these
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*> > are known as non-Cantorian set theories.
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*> >
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*> > On the other hand, in standard set theory, assuming the CH does no
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*> > harm.
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*> >
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*> > [More below]
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*> >
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*> > >
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*> > > This is pretty much over my head. As I understand it aleph 1 is
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*> > > defined to be the next cardinal number after aleph 0, and can be
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*> > > shown to be the cardinality of the set of all countable ordinals
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*> > > (1...w...w+1...2w...w^2...). Since the elements of this set are all
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*> > > ordered sets (i.e. ordinals), while the subsets of N don't have an
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*> > > ordering requirement, this gives more flexibility to c and so you can't
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*> > > compare them as simply as you have shown here.
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*> > >
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*> > > See http://www.ii.com/math/ch/ for a detailed discussion of these
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*> > > matters.
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*> > >
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*> > > Actually on further thought I think I was wrong to suggest that the
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*> > number
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*> > > of TM programs is c, since that would allow for infinite length
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*> > programs,
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*> > > which is perhaps outside the spirit of a TM. If we require only finite
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*> > > length programs then the number of TMs is aleph 0 since we can enumerate
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*> > > all the programs, and that would be the number of universes as well.
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*> > > Not so many after all.
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*> > >
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*> > > Hal
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*> >
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*> > I think you are right that the cardinality of the set of all programs
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*> > is aleph-0.
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*> >
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*> > But neither of you (nor anyone else) has addressed my reason for
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*> > conjecturing that the set of branches in our structure must be
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*> > aleph-(aleph-0), which is based on the SSA.
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*> >
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*> > To tell you the truth, I'm certainly not convinced of it, but I
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*> > think it's worth considering. To discard the conclusion, I would
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*> > think that you'd have to assume "the identity of indiscernables".
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*> > My reasoning is illustrated if you only assume, for the moment,
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*> > that some observable (say, x) can take a continuum of possible
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*> > values when measured. Forget about the Plank length, for now.
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*> > That would mean that the set of all possible humans would have
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*> > cardinality c (at least). Thus it would be impossible to map
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*> > that set onto the set of all programs.
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*> >
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*> > But if you believe in the computationalist hypothesis, then you'd
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*> > have to assume that at some point, a simulation of a human becomes
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*> > "close enough" to be identical. That is (to oversimplify) when
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*> > each particle is simulated to within a Plank length, then the
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*> > simulation becomes indiscernable from the original, and thus
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*> > identical. If this is true, then I would no longer expect the
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*> > physical laws to give rise to ever-increasing cardinality of
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*> > universes, since that could never increase the cardinality of
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*> > the set of humans past aleph-0, anyway.
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*> >
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*> >
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*> >
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*> > --
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*> > Chris Maloney
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*> > http://www.chrismaloney.com
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*> >
*

*> > "Knowledge is good"
*

*> > -- Emil Faber
*

Date: Fri, 16 Jul 1999 23:33:26 -0400

Higgo James wrote:

Umm, no! "Space" is an emergent phenomenon as well. What could you

possibly mean that it has a different "spatial coordinate"?

-- Chris Maloney http://www.chrismaloney.com "Knowledge is good" -- Emil FaberReceived on Fri Jul 16 1999 - 20:57:58 PDT

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